Chap.6 The Impulse-Momentum Principle

Chap.6 The Impulse-Momentum Principle

• So far, we have studied the flow characteristics (velocity, pressure, etc.) in a fluid. In this chapter, we study the force acting on a structure by fluid in a flow. As for the force acting on a structure in quiescent fluid, we studied it in Chapter 2.

• Newton’s 2nd law (in a vector form):

( )

^c

c mv

dt a d

m

F = =

∑

where v_c = velocity of the center of mass of the system

( ) ( )

₁₂₃

43 42 1

momentum linear time

the in impulse

c

dt

v m d dt F =

∑

• Linear impulse-momentum equation

→ magnitude and direction of resultant forces exerted on fluid by structures (or on structures by fluid)

• Angular impulse-momentum equation

→ line of action of the resultant force

6.1 Linear Impulse-Momentum Equation

For the individual fluid system,

( ) (

^vdV

)

dt v d

dt m a d

m

F = = = ρ

∑

Summation over the whole flow system:

( )

(

^{2 1}

)

1 1 1 2 2

2

CS CS

system system

ext

in out

) (

v )

( v

V V

Q Q

V Q

V

dA n v dA

n v

V d dt v

V d d dt v

F d

−

=

−

=

⋅ +

⋅

=

=

=

∫∫

∫∫

∫∫∫

∑ ∫∫∫

ρ ρ

ρ

ρ ρ

ρ ρ

where

∑

^Fext = body force + forces on C.S.

(Q all the internal forces b/w individual fluid systems are cancelled out)

(

^{2 1}

)

ext Q V V

F = −

∴

∑

^ρ

In a 2-D flow (_x and z),

( ) ( )

( ) ( )

^⎪_⎭

⎪⎬

⎫

−

=

−

=

∑

∑

z z

x x

V V

Q F

V V

Q F

z x

1 2

ext

1 2

ext

ρ ρ

Magnitude and direction of

∑

^Fext.

But line of action is not known yet.

If the flow enters and leaves at more than one location,

( ) (

_out

)

_in

ext

∑ ∑

∑

^{F = QρV − QρV}

6.2 Pipe Flow Applications

(1) Force acting on pipe bend

) cos

( cos

: p₁A₁ p₂A₂ F Q V₂ V₁

x − α − _x = ρ α −

_{Fx = p1A1 − p2A2cos}α _−Qρ_{(V2 cos}α _−V1) ) 0 sin

( sin

: −W − p₂A₂ α + F =Qρ V₂ α −

z _z

F_z =W + p₂A₂sinα +QρV₂sinα

(2) Energy loss at abrupt expansion of a passage

Impulse-momentum equation on C.V. ABCD:

) (

)

( _{2 1 2 2 2 1}

2 2 2

1A p A Q V V AV V V

p

F_x ≈ − = − = −

∑

^{ρ ρ}

Note: _p may be different from _p1 in eddy area.

g

V V V p

p₁ − ₂ = ₂( ₂ − ₁)

∴ γ

Work-energy equation:

g V p

g V p

(EL) 2 2

2 2 2

2 1

1 + −Δ = +

γ γ

g V V

g V V V g

V V

p p

2 )

( (EL) 2

2 2 2 1 1

2 2 2

2 2 1 2

1 − + − = − + −

= Δ

∴ γ

g V V

2 ) ( ₁ − ₂ ²

= : Borda-Carnot head loss

6.3 Open Channel Flow Applications

(1) Force acting on a sluice gate

( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

=

−

=

−

−

∑

=

1 2

1 2 2

2 2

1

ext ( )

2

2 y

q y

q q V

V Q y F

F y _x

x γ γ ρ ρ

( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

−

−

=

∴

1 2

2 2

2 2

1

1 1

2 y y q y y

F_x γ ρ

⊥

F gate (Q no shear force for ideal fluid) Fx

F =

∴ cosθ →

θ cos

Fx

F =

θ sin F F_z =

No need for impulse-momentum equation in z- direction.

See IP 6.2 (p 197) for horizontal force acting on an overflow structure (same principle).

(2) Energy loss due to hydraulic jump

Impulse-momentum equation:

( )

^{( )}

2

2 ^{2 1}

2 2 2

1

ext y y q V V

F x = − = −

∑

^{γ γ ρ ←} no structure

no external force by structure Continuity: _{q =V1y1 =V2y2} →

2 2

1

1 ,

y V q

y

V = q =

( )

_⎟⎟

⎠

⎜⎜ ⎞

⎝

⎛ −

=

−

∴

1 2

2 2

2 2

1

1 1

2g y y q ρ y y

ρ

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ −

=

−

2 1

2 2

1 2 2

1 1

2

y y

g y q

y

( )( )

2 1

1 2 2 1

2 1 2

2

y y

y y

g y q

y y

y + − = −

2 0

3 1

2

1 2 2

1

2 ⎟⎟⎠ + − =

⎜⎜ ⎞

⎝

⎛

gy q y

y y

y

⎟⎟

⎠

⎞

⎜⎜

⎝

⎛− + +

⎟ =

⎟

⎠

⎞

⎜⎜

⎝

⎛− + +

=

∴

1 2 1 3

1 2

1

2 8

1 2 1

1 1 8

2 1 1

gy V gy

q y

y

(a) 1

1 2 2 1

1 = =

gy

F V → 1

1 2 = y

y ; no hydraulic jump

(b) 1

1 2 2 1

1 = >

gy

F V → 1

1 2 >

y

y ; hydraulic jump occurs

(c) 1

1 2 2 1

1 = <

gy

F V → 1

1 2 <

y

y ; physically impossible Note: = = Froudenumber

gy F V

⎟⎟⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟−

⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ =

⎠

⎜⎜ ⎞

⎝

⎛ +

⎟⎟ −

⎠

⎜⎜ ⎞

⎝

⎛ +

=

Δ ₂

2 2 2 2

1 2 1

2 2 2

2 1

1 2 2 2 2

) EL

( gy

y q gy

y q g

y V g

y V

Using _{( )}

2 ^{1 2 1 2}

2 g y y y y

q = +

( )

4 0 EL)

(

2 1

3 1 2 − >

=

Δ y y

y

y for energy loss

1

2 y

y >

∴ → 1

1 2 >

y y

(3) Velocity of small gravity waves

not fluid velocity but propagating speed of waves

Under the wave, fluid velocity is unknown except that it is constant over depth for a small gravity wave (or long wave).

For a stationary observer, it is an unsteady problem. For an observer moving with speed _a, however, it is a steady problem.

Continuity: ay =V(y + dy) = q

Impulse-momentum: _{( )}

2 ) (

2

2 2

a V dy q

y

y + = −

− γ ρ

γ

⎟⎠

⎜ ⎞

⎝⎛ +

⎟⎠

⎜ ⎞

⎝

⎛ +

=

∴ y dy

y g dy

a 2

1 1

2

If y >> dy, then _{a ≈ gy} ← a depends only on y.

6.4 Angular Impulse-Momentum Principle

line of action of resultant forces

( ∑F = ma)moment→ moment of momentum

= angular momentum

( ) ( ) (

^{r v dV}

)

dt v d

m dt r

a d m r F

r× = × = × = × ρ

∑

Summation over the whole flow system:

( ) ∫∫∫ ( )

∑

^{× = ×}

sys

ext r v dV

dt F d

r ρ

( ) ( ) ( ) ( ∫∫ )

∫∫

^{× ⋅ + × ⋅}

=

in

out CS

CS

dA n v v r dA

n v v

r ρ ρ

( ) ∫∫ ( )

∫∫

^{× − ×}

=

in

out CS

CS

dQ v

r dQ

v

r ρ ρ

(

^{rout Vout}

)

^Q

(

^{rin Vin}

)

Q × − ×

= ρ ρ

( ) ( ) [

r V out r V in

]

Q × − ×

= ρ

( ) ∑ [ ( ) ( ) ]

∑

^{× = = × − ×}

∴ r Fext M 0 Qρ r V _out r V _in

Note: _{r = xi + y j + zk}, V =V_xi +V_y j +V_zk

(

^{yV zV}

)

ⁱ

(

^{zV xV}

)

^j

(

^{xV yV}

)

^k

V V

V

z y x

k j i

V

r _{z y x z y x}

z y x

− +

− +

−

=

=

×

In 2-D problem (_x,z),

(

zV xV

)

j V

V

z x

k j i V

r _{x z}

z x

−

=

=

×

0 0

Similarly, ^r×^F =

(

^zF_x − ^xF_z

)

^j

( ) [ ( ) ( ) ]

∑

^{− = − − −}

∴ zF_extx xF_extz Qρ z₂V₂x x₂V₂z z₁V₁x x₁V₁z

where 1=CS_in and 2=CS_out.

For the resultant force, use _{Fext ×r} , where _Fext is known from linear impulse-momentum equation. For other external forces (i.e. body force and pressure forces at 1 and 2), we know the point of action (i.e. _x and z).